Generalized Model Predictive Static Programming and Angle-Constrained Guidance of Air-to-Ground Missiles

A new generalized model predictive static programming technique is presented for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. Two key features for its high computational efficiency include one-time backward integration of a small-dimensional weighting matrix dynamics, followed bya static optimization formulation that requires only a static Lagrange multiplier to update the control history. It turns out that under Euler integration and rectangular approximation of finite integrals it is equivalent to the existing model predictive static programming technique. In addition to the benchmark double integrator problem, usefulness of the proposed technique is demonstrated by solving a three-dimensional angle-constrained guidance problem for an air-to-ground missile, which demands that the missile must meet constraints on both azimuth and elevation angles at the impact point in addition to achieving near-zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Simulation studies include maneuvering ground targets along with a first-order autopilot lag. Comparison studies with classical augmented proportional navigation guidance and modern general explicit guidance lead to the conclusion that the proposed guidance is superior to both and has a larger capture region as well.

Capturability Analysis of a 3-D Retro-PN Guidance Law for Higher Speed Nonmaneuvering Targets

This brief presents the capturability analysis of a 3-D Retro-proportional navigation (Retro-PN) guidance law, which uses a negative navigation constant (as against the usual positive one), for intercepting targets having higher speeds than interceptors. This modification makes it possible to achieve collision conditions that were inaccessible to the standard PN law. A modified polar coordinate system, that makes the model more compact, is used in this brief for capturability analysis. In addition to the ratio of the target to interceptor speeds, the directional cosines of the interceptor, and target velocity vectors play a crucial role in the capturability. The existence of nontrivial capture zone of the Retro-PN guidance law and necessary and sufficient conditions, for capturing the target in finite time, are presented. A sufficient condition on the navigation constant is derived to ensure finiteness of the line-of-sight turn rate. The results are more extensive than those available for 2-D engagements, which can be obtained as special cases of this brief. Simulation results are given to support the analytical results.

Formation Flying with Nonlinear Partial Integrated Guidance and Control

Using the dynamic inversion philosophy, a nonlinear partial integrated guidance and control approach is presented in this paper for formation flying. It is based on the evolving philosophy of integrated guidance and control. However, it also retains the advantages of the conventional guidance then control philosophy by retaining the timescale separation between translational and rotational dynamics explicitly. Simulation studies demonstrate that the proposed technique is effective in bringing the vehicles into formation quickly and maintaining the formation.

Path Following Using Trajectory Shaping Guidance

Unmanned vehicle path following by pursuing a virtual target moving along the path is considered. Limitations for pure pursuit guidance are analyzed while following the virtual target on curved paths. Trajectory shaping guidance is proposed as an alternate guidance scheme for a general curvature path. It is proven that under certain tenable assumptions trajectory shaping guidance yields an identical path as that of the virtual target. By linear analysis it is shown that the convergence to the path for trajectory shaping guidance is twice as fast as pure pursuit. Simulations highlight significant improvement in position errors by using trajectory shaping guidance. Comparative simulation studies comply with analytic findings and present better performance as compared with pure pursuit and a nonlinear guidance methodology from the literature. Experimental validation supports the analytic and simulations studies as the guidance laws are implemented on a radio-controlled car in a laboratory environment.

Three Dimensional Impact Angle Constrained Guidance Law using Sliding Mode Control

In this paper, three dimensional impact angle control guidance laws are proposed for stationary targets. Unlike the usual approach of decoupling the engagement dynamics into two mutually orthogonal 2-dimensional planes, the guidance laws are derived using the coupled dynamics. These guidance laws are designed using principles of conventional as well as nonsingular terminal sliding mode control theory. The guidance law based on nonsingular terminal sliding mode guarantees finite time convergence of interceptor to the desired impact angle. In order to derive the guidance laws, multi-dimension switching surfaces are used. The stability of the system, with selected switching surfaces, is demonstrated using Lyapunov stability theory. Numerical simulation results are presented to validate the proposed guidance law.

An integrated estimation/guidance approach for seeker-less interceptors

This paper addresses the problem of intercepting highly maneuverable threats using seeker-less interceptors that operate in the command guidance mode. These systems are more prone to estimation errors than standard seeker-based systems. In this paper, an integrated estimation/guidance (IEG) algorithm, which combines interactive multiple model (IMM) estimator with differential game guidance law (DGL), is proposed for seeker-less interception. In this interception scenario, the target performs an evasive bang-bang maneuver, while the sensor has noisy measurements and the interceptor is subject to acceleration bound. The IMM serves as a basis for the synthesis of efficient filters for tracking maneuvering targets and reducing estimation errors. The proposed game-based guidance law for two-dimensional interception, later extended to three-dimensional interception scenarios, is used to improve the endgame performance of the command-guided seeker-less interceptor. The IMM scheme and an optimal selection of filters, to cater to various maneuvers that are expected during the endgame, are also described. Furthermore, a chatter removal algorithm is introduced, thus modifying the differential game guidance law (modified DGL). A comparison between modified DGL guidance law and conventional proportional navigation guidance law demonstrates significant improvement in miss distance in a pursuer-evader scenario. Simulation results are also presented for varying flight path angle errors. A numerical study is provided which demonstrates the performance of the combined interactive multiple model with game-based guidance law (IMM/DGL). Simulation study is also carried out for combined IMM and modified DGL (IMM/modified DGL) which exhibits the superior performance and viability of the algorithm reducing the chattering phenomenon. The results are illustrated by an extensive Monte Carlo simulation study in the presence of estimation errors.

Nonlinear Differential Games-Based Impact-Angle-Constrained Guidance Law

The problem of intercepting a maneuvering target at a prespecified impact angle is posed in nonlinear zero-sum differential games framework. A feedback form solution is proposed by extending state-dependent Riccati equation method to nonlinear zero-sum differential games. An analytic solution is obtained for the state-dependent Riccati equation corresponding to the impact-angle-constrained guidance problem. The impact-angle-constrained guidance law is derived using the states line-of-sight rate and projected terminal impact angle error. Local asymptotic stability conditions for the closed-loop system corresponding to these states are studied. Time-to-go estimation is not explicitly required to derive and implement the proposed guidance law. Performance of the proposed guidance law is validated using two-dimensional simulation of the relative nonlinear kinematics as well as a thrust-driven realistic interceptor model.

Nonlinear control of an air-breathing engine including its validation with vehicle guidance

An implementable nonlinear control design approach is presented for a supersonic air-breathing ramjet engine. The primary objective is to ensure that the thrust generated by the engine tracks the commanded thrust without violating the operational constraints. An important constraint is to manage the shock wave location in the intake so that it neither gets detached nor gets too much inside the intake. Both the objectives are achieved by regulating the fuel flow to the combustion chamber and by varying the throat area of the nozzle simultaneously. The design approach accounts for the nonlinear cross-coupling effects and nullifies those. Also, an extended Kalman filter has been used to filter out the sensor and process noises as well as to make the states available for feedback. Furthermore, independent control design has been carried out for the actuators. To test the performance of the engine for a realistic flight trajectory, a representative trajectory is generated through a trajectory optimization process, which is augmented with a newly-developed finite-time state dependent Riccati equation technique for nullifying the perturbations online. Satisfactory overall performance has been obtained during both climb and cruise phases. (C) 2015 Elsevier Masson SAS. All rights reserved.

Impact Time Guidance for Large Heading Errors Using Sliding Mode Control

In this paper, sliding mode control-based impact time guidance laws are proposed. Even for large heading angle errors and negative initial closing speeds, the desired impact time is achieved by enforcing a sliding mode on a switching surface designed by using the concepts of collision course and estimated time-to-go. Unlike existing guidance laws, the proposed guidance strategy achieves impact time successfully even when the estimated interception time is greater than the desired impact time. Simulation results are also presented.

Selection of Optimal Time-to-go in Generalized Vector Explicit Guidance

A new method of selection of time-to-go (t(go)) for Generalized Vector Explicit Guidance (GENEX) law have been proposed in this paper. t(go) is known to be an important parameter in the control and cost function of GENEX guidance law. In this paper the formulation has been done to find an optimal value of t(go) that minimizes the performance cost. Mechanization of GENEX with this optimal t(go) reduces the lateral acceleration demand and consequently increases the range of the interceptor. This new formulation of computing t(go) comes in closed form and thus it can be implemented onboard. This new formulation is applied in the terminal phase of an surface-to-air interceptor for an angle constrained engagement. Results generated by simulation justify the use of optimal t(go).

Inverse Polynomial Based Explicit Guidance for Lunar Soft Landing During Powered Braking

In this paper an explicit guidance law for the powered descent phase of the soft lunar landing is presented. The descent trajectory, expressed in polynomial form is fixed based on the boundary conditions imposed by the precise soft landing mission. Adapting an inverse model based approach, the guidance command is computed from the known spacecraft trajectory. The guidance formulation ensures the vertical orientation of the spacecraft during touchdown. Also a closed form relation for the final flight time is proposed. The final time is expressed as a function of initial position and velocity of the spacecraft ( at the start of descent) and also depends on the desired landing site. To ensure the fuel minimum descent the proposed explicit method is extended to optimal guidance formulation. The effectiveness of the proposed guidance laws are demonstrated with simulation results.

Fuel-optimal G-MPSP Guidance for Powered Descent Phase of Soft Lunar Landing

A fuel optimal nonlinear sub-optimal guidance scheme is presented in this paper for soft landing of a lunar craft during the powered descent phase. The recently developed Generalized Model Predictive Static Programming (G-MPSP) is used to compute the required magnitude and angle of the thrust vector. Both terminal position and velocity vector are imposed as hard constraints, which ensures high position accuracy and facilitates initiation of vertical descent at the end of the powered descent phase. A key feature of the G-MPSP algorithm is that it converts the nonlinear dynamic programming problem into a low-dimensional static optimization problem (of the same dimension as the output vector). The control history update is done in closed form after computing a time-varying weighting matrix through a backward integration process. This feature makes the algorithm computationally efficient, which makes it suitable for on-board applications. The effectiveness of the proposed guidance algorithm is demonstrated through promising simulation results.

Composite Guidance for Impact Angle Control Against Higher Speed Targets

In the literature, the impact angle control problem has been addressed mostly against lower speed or stationary targets. However, in the current defense scenario, targets of much higher speeds than interceptors are a reality. Moreover, approaching a higher speed target from a specified angle is important for effective seeker acquisition and enhanced warhead effectiveness. This paper proposes a composite proportional navigation guidance law using a combination of the standard proportional navigation and the recently proposed retroproportional navigation guidance laws for intercepting higher speed nonmaneuvering targets at specified impact angles in three-dimensional engagements. An analysis of the set of achievable impact angles by the composite proportional navigation guidance law is presented. It is shown that there exists an impulse bias that, when added to the composite proportional navigation guidance command, expands this set further by reversing the direction of the line-of-sight angular rotation vector. A bound on the magnitude of the bias is also derived. Finally, an implementation of this impulse bias, in the form of a series of pulses, is proposed and analyzed. Simulation results are also presented to support the analysis.